Method and/or system for transforming between trees and strings

ABSTRACT

Embodiments of articles, apparatuses, devices and/or systems for manipulating hierarchical sets of data are disclosed.

RELATED APPLICATION

This application claims priority to U.S. Provisional Patent Application Ser. No. 60/657,264 filed Feb. 28, 2005, entitled METHOD AND/OR SYSTEM FOR TRANSFORMING BETWEEN TREES AND STRINGS, assigned to the assignee of claimed subject matter.

BACKGROUND

This disclosure is related to hierarchical data arrangements and, more particularly, to manipulating such data arrangements.

In a variety of fields, data or a set of data, may be represented in a hierarchical fashion. This form of representation may, for example, convey information, such as particular relationships between particular pieces of data and the like. However, manipulating such data representations is not straight-forward, particularly where the data is arranged in a complex hierarchy. Without loss of generality, one example may include a relational database. Techniques for performing operations on such a database, for example, are computationally complex or otherwise cumbersome. A continuing need, therefore, exists for additional techniques for manipulating data hierarchies.

BRIEF DESCRIPTION OF THE DRAWINGS

Subject matter is particularly pointed out and distinctly claimed in the concluding portion of the specification. The claimed subject matter, however, both as to organization and method of operation, together with objects, features, and advantages thereof, may best be understood by reference of the following detailed description when read with the accompanying drawings in which:

FIG. 1 is a schematic diagram of an unordered tree;

FIG. 2 is a table illustrating an embodiment of a relationship between binary edge labeled trees (BELTs) and positive natural numerals;

FIG. 3 is a table providing an embodiment of a function that relates natural numerals to non-composite numerals;

FIG. 4 is a schematic diagram illustrating an embodiment of an unordered edge-labeled tree and a symbolic expression mathematically representing the tree embodiment;

FIG. 5 is a schematic diagram of embodiments of unordered edge-labeled trees and symbolic expressions mathematically representing the tree embodiments;

FIG. 6 is a schematic diagram illustrating all N-valued rooted, unordered, edge-labeled trees, where N is a natural numeral greater than or equal to two;

FIGS. 7 through 9 are tables illustrating embodiments of a relationship among symbol strings, trees and natural numerals;

FIG. 10 is a table illustrating an embodiment of a relationship among natural numerals, BELTs and symbol strings comprising two types of grouping symbol pairs;

FIG. 11 is a table illustrating an embodiment of a relationship between natural numerals, tertiary A, B, C, edge labeled trees and symbol strings comprising three types of grouping symbol pairs;

FIGS. 12 and 13 are schematic diagrams of embodiments of relationship between BELTs and symbol strings representing the BELT embodiments; and

FIGS. 14 and 15 are schematic diagrams of embodiments of edge labeled trees and symbol strings representing the tree embodiments.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are set forth to provide a thorough understanding of the claimed subject matter. However, it will be understood by those skilled in the art that the claimed subject matter may be practiced without these specific details. In other instances, well-known methods, procedures, components and/or circuits have not been described in detail so as not to obscure the claimed subject matter.

Some portions of the detailed description which follow are presented in terms of algorithms and/or symbolic representations of operations on data bits or binary digital signals stored within a computing system, such as within a computer or computing system memory. These algorithmic descriptions and/or representations are the techniques used by those of ordinary skill in the data processing arts to convey the substance of their work to others skilled in the art. An algorithm is here, and generally, considered to be a self-consistent sequence of operations and/or similar processing leading to a desired result. The operations and/or processing involve physical manipulations of physical quantities. Typically, although not necessarily, these quantities may take the form of electrical and/or magnetic signals capable of being stored, transferred, combined, compared and/or otherwise manipulated. It has proven convenient, at times, principally for reasons of common usage, to refer to these signals as bits, data, values, elements, symbols, characters, terms, numbers, numerals and/or the like. It should be understood, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels. Unless specifically stated otherwise, as apparent from the following discussion, it is appreciated that throughout this specification discussions utilizing terms such as “processing”, “computing”, “calculating”, “transforming,” “determining” and/or the like refer to the actions and/or processes of a computing platform, such as a computer or a similar electronic computing device, that manipulates and/or transforms data represented as physical electronic and/or magnetic quantities and/or other physical quantities within the computing platform's processors, memories, registers, and/or other information storage, transmission, and/or display devices.

In a variety of fields, data or a set of data, may be represented in a hierarchical fashion. This form of representation may, for example, convey information, such as particular relationships or patterns between particular pieces of data or groups of data and the like. However, manipulating and/or even recognizing specific data representations or patterns is not straight-forward, particularly where the data is arranged in a complex hierarchy. Without loss of generality, examples may include a database and further, without limitation, a relational database. Techniques for performing operations on such databases or recognizing specific patterns, for example, are computationally complex, time consuming, and/or otherwise cumbersome. A need, therefore, continues to exist for improved techniques for performing such operations and/or recognizing such patterns.

As previously discussed, in a variety of fields, it is convenient and/or desirable to represent data, a set of data and/or other information in a hierarchical fashion. In this context, such a hierarchy of data shall be referred to as a “tree.” In a particular embodiment, a tree may comprise a finite, non-empty (including at least one node), rooted, connected, acyclic graph. Likewise, such trees may be either ordered or unordered. Further discussion of non-empty trees may be found in The Art of Computer Programming, Vol. 1, Fundamental Algorithms, Donald Knuth, Addison Wesley. Here, ordered refers to the notion that there is an ordering or precedence among nodes attached to a common node corresponding to the order of the attached nodes shown in a graphical illustration. An unordered tree is illustrated here, for example, in FIG. 1 by embodiment 100. As illustrated, the root of this particular embodiment encompasses node 105. In addition to 105, there are eight other nodes designated 110 to 145, respectively. Likewise, the nodes are connected by branches referred to, in this context, as edges. Thus, the nodes of this tree are connected by eight edges. This embodiment, therefore, illustrates a finite tree that is rooted by node 105. Furthermore, the nodes are connected, meaning, in this context, that a path exists between any two nodes of the tree. The tree is likewise acyclic, meaning here, that no path in the tree forms a complete non-backtracking loop.

As previously suggested, in a variety of contexts, it may be convenient and/or desirable to represent a hierarchy of data and/or other information using a structure, such as the embodiment illustrated in FIG. 1. One particular embodiment, without loss of generality, of a tree may include edges that are labeled with data and/or other values represented as “edge label values.” Similarly, in another particular embodiment, a tree may include nodes that are labeled with data and/or values represented as “node label values.” Likewise, in one particular embodiment, such data and/or values may be limited to a particular set of data. For example, in this context, a binary edge labeled tree refers to a tree in which the data and/or values comprise binary data, that is, in this example, either a binary one or a binary zero. Likewise, alternatively, the edges of a tree may be labeled with three values, such as 0, 1, 2. Continuing, the edges may be labeled with four values, five values, etc. In this context, the class of all trees in which the edges are labeled with a specific number of distinct values, that is, in this context, values chosen from a set having a specific number of distinct elements, shall be referred to as edge-labeled trees (ELTs). It is likewise noted that such trees are not limited to being labeled with the numerals previously described. Any distinctly identifiable labels may be employed; however, in this context, it shall be understood that employing numerals to label the edges is sufficiently general to encompass any sort of data labels that may be desirable, regardless of their form.

To reiterate, in this context, a tree comprises an edge labeled tree if each edge of the tree respectively holds, stores and/or represents a value or piece of data. Likewise, in this context, two nodes are employed to support an edge storing, holding and/or representing a piece of data. At this point, it is worth noting that trees having nodes and edges, such as previously described, may be represented in a computing platform or similar computing device through a data structure or a similar mechanism intended to capture the hierarchical relationship of the data, for example. It is intended that all such embodiments are included within the scope of the claimed subject matter.

According to an embodiment, trees or portions of trees may be represented by natural numerals according to an association of trees and natural numerals. Without belaboring the present discussion, additional descriptions of how natural numerals may be associated with trees are discussed in greater detail below and in U.S. patent application Ser. No. 11/005,859, filed on Dec. 6, 2004, by J. J. LeTourneau, titled, “Manipulating Sets of Hierarchical Data,” assigned to the assignee of the presently claimed subject matter. It should be understood, however, that while the particular embodiments illustrated are directed to particular associations of numerals and trees, there may be many associations of trees to numerals according to corresponding particular “association embodiments,” and the claimed subject matter is not limited to any such particular association.

According to an embodiment, although the claimed subject matter is not limited in scope in this respect, a method of enumerating a set of unordered trees may begin with enumeration of a one node binary edge labeled tree. The one node tree, which holds no data, is associated with the natural numeral one and has a graphical representation of a single node in this embodiment. For higher natural numbers, ordered trees may be generated by a process described, for example, in “The Lexicographic Generation of Ordered Trees,” by S. Zaks, The Journal of Theoretical Computer Science, Vol. 10(1), pp63-82, 1980, or, “Enumerating Ordered Trees Lexicographically,” by M. C. Er, Computation Journal, Vol. 28, Issue 5, pp 538-542, 1985.

As illustrated, for this particular embodiment, the one node tree root comprises a single node and is associated with the natural numeral one. Thus, to obtain the tree at position two, a root node is attached and connected to the prior root node by an edge. Likewise, here, by convention, the edge is labeled with a binary zero. If, however, the tree formed by the immediately proceeding approach were present in the prior enumeration of trees, then a similar process embodiment is followed, but, instead, the new edge is labeled with a binary one rather than a binary zero. Thus, for example, to obtain the binary edge labeled tree for position three, a new root node is connected to the root node by an edge and that edge is labeled with a binary one. It should be understood that, while this particular embodiment associates a single node tree with the numeral one, and associates other trees with other numerals based, at least in part, upon this association, other embodiments illustrated below associate a single node tree with the numeral zero, or different numerals. Other trees in this association embodiment may then be based, at least in part, on this association.

Continuing with this example, to obtain the binary edge labeled tree for position four, observe that numeral four is the product of numeral two times numeral two. Thus, a union is formed at the root of two trees, where, here, each of those trees is associated with the natural numeral two. Likewise, to obtain the binary edge labeled tree for position five, begin with the binary edge labeled tree for position two and follow the previously articulated approach of adding a root and an edge and labeling it with a binary zero.

In this context, adding a root node and an edge to a BELT and labeling it binary zero is referred to as a “zero-push” operation and adding a root node and an edge to the BELT and labeling it binary one is referred to as a “one-push” operation. Here, a numeral associated with a tree resulting from either a zero-push or a one-push operation, according to a particular association embodiment, may be represented as a numerical operation on the numeral representing the BELT as follows: zero-Push(x)=Q(2x−2) one-Push(x)=Q(2x−1)

-   -   where:     -   x=numeral associated with BELT according to the particular         association embodiment; and     -   Q(k) is a function generating an ordered sequence of         non-composite numerals as shown in FIG. 3.

Thus, referring again to FIG. 2, the one-push of the root tree is the tree at position three. This follows since Q((1*2)−1)=Q(1)=3. Likewise, the tree at position five is the zero-push of the tree at position 2. This follows since Q((2*2)−2)=Q(2)=5.

In the presently illustrated embodiment, the function Q(k) relates to the well known Kleene enumeration function for the sequential enumeration of non-composite numerals P(k). Accordingly, in this context, the term prime numerals and non-composite numerals may be used interchangeably throughout. Here, the function Q(k) may be generated as Q(k)=P(k+1) where the Kleene enumeration function is expressed, in part, as follows: P(1)=2 P(2)=3 P(3)=5 P(4)=7 . . .

In the embodiment just described, binary edge labeled trees use binary numerals “0” and “1.” However, the claimed subject matter is not limited in scope to binary edge labeled trees. For example, trees may employ any number of numeral combinations as labels, such as triplets, quadruplets, etc. Thus, using a quadruplet example, it is possible to construct trees, such as a zero-push of a particular tree, a one-push of that tree, a two-push of that tree, and a three-push of that tree.

The foregoing discussion has begun to characterize an algebra involving trees, in this particular embodiment, an algebra for unordered edge labeled trees or unordered ELTs, such as BELTs. The foregoing discussion define a value one, a one node tree for this particular embodiment, and a monadic operation, previously described as zero-push. For example, alternatively, a “one-push” may be employed. For this embodiment, this is analogous, for example, to the convention that “0” represent “off” and “1,” represent “on.” Alternatively and equivalently, “1” may be employed to represent “off,” and “0” may be employed to represent “on,” without loss of generality. For this particular embodiment, an additional operation may be characterized, a “merger” operation. The merger operation with respect to trees refers to merging two trees at their roots.

As will now be appreciated, the merger operation comprises a dyadic operator or, in this context, also known as a binary operator. Likewise, the constants zero/one, referred to above, may be viewed as an operation having no argument or as a zero valued argument operator or operation. Thus, this operation, in effect, returns the same value whenever applied. Here, for this particular embodiment, the constant value, or zero valued argument operation that returns “c” and is denoted as “c.” The merger operator is denoted as “*”.

FIG. 4 is schematic diagram illustrating an embodiment of an edge labeled tree, here a four valued edge labeled tree. In this particular embodiment, four distinct values are employed to label the edges. Here, the labels comprising A, B, C and D, although, of course, the claimed subject matter is not limited to four valued edge labeled trees, to edge labeled trees, or to employing these particular edge labels. It is noted that the labels A, B, C, and D in this embodiment are similar to the labels binary 0 and binary 1 for BELTs. Below tree 400 in FIG. 4 is a symbolic expression mathematically representing tree 400. Performing the operations indicated by the expression shown in FIG. 4 below tree 400 will provide a natural numeral that corresponds, for this particular embodiment, to this particular tree, as described in more detail hereinafter.

To assist in understanding the relationship between the symbolic expression shown in FIG. 4 and tree 400, for this particular embodiment, FIG. 5 provides an embodiment 510 of another tree. As illustrated, tree 510 comprises an edge label D connecting two nodes. For this particular context, this embodiment may be expressed symbolically as follows: D(1). Thus, a technique to describe the embodiment of tree 510 would refer to the “push” of the natural numeral 1. Here, for this particular embodiment, this particular push operation comprises the “D” push of 1, resulting in D being the label of the edge connecting the two nodes. More specifically, as previously described, a single node is associated with the natural numeral 1 in this particular embodiment. To perform a push operation, an edge is attached to that node and labeled. Here, applying a D push, the label provided comprises the label D.

Continuing, the “C” push of “1” is illustrated as two nodes with an edge labeled C connecting the two nodes for tree embodiment 520. Applying similar reasoning provides an edge labeled tree embodiment 530 representing the following expression: B(C(1)). Likewise, for this particular embodiment, the operation of merger may be represented as “*”, as previously suggested. Thus, applying a merger operation provides tree embodiment 540 at the bottom of FIG. 5 corresponding, for this particular embodiment, to the following expression: (D(1)*B(C(1))).

Previously, an embodiment for manipulating binary edge labeled trees or BELTs was described in connection with the aforementioned U.S. patent application Ser. No. 11/005,859. In that context, binary edge labeled trees comprise finite rooted, unordered two valued edge labeled trees. Thus, for the particular embodiment of binary edge labeled trees described, the two values comprise “0” and “1,” although alternately they could comprise A and B, for example, or any other two values. Referring now to FIG. 6, a Venn diagram 600 is illustrated providing the set of all edge labeled trees, structured or organized in a particular manner here. In the center of the diagram, binary or two valued edge labeled trees are depicted as a subset. Furthermore, as illustrated, two valued edge labeled trees are also depicted as a subclass or subset of tertiary or three valued edge labeled trees. Likewise, three valued edge labeled trees are depicted as a subclass or subset of four valued edge labeled trees and so forth. Thus, depending at least in part on the particular set of distinct values employed to label the edges, an edge labeled tree that employs two distinct values may comprise an example of a three valued edge labeled tree in which one of the values is specifically not employed in the particular tree. As shall be explained in more detail hereinafter, this raises a question regarding proper interpretation of the data that the tree may hold, represent and/or store. More specifically, an identical tree may represent different data depending at least in part on whether the tree is “viewed” as, to continue with this example, a two valued edge labeled tree or a three valued edge labeled tree. For example, up to two views of a two valued tree may be obtained and up to three views of a three valued tree may be obtained.

Thus, in this embodiment, we refer to this as the “view” of the particular expression. While particular embodiments illustrated herein relate to obtaining views of particular expressions known as trees, other embodiments may relate to obtaining views of other types of expressions such as symbol strings as illustrated below. However, symbol strings and trees are merely examples of expressions from which a view may be obtained and claimed subject matter is not limited in this respect. For a particular example of expressions comprising trees, for the purposes of illustration, a two valued edge labeled tree is referred to as view 2 and a three valued edge labeled tree is referred to as view 3, although, for example, the particular tree may not contain three different values. The view in this embodiment is determined, at least in part, by the set of distinct values from which the labels may be selected, as previously described. FIG. 6 therefore depicts the set of all edge labeled trees as the union of all such edge labeled trees in which the edge values are selected from a set having a specific number of distinct values.

Previously in the aforementioned U.S. patent application Ser. No. 11/005,859, an embodiment was demonstrated in which an association existed between natural numerals and binary edge labeled trees. For this particular embodiment, similar associations also exist, here between any N valued edge labeled tree and the natural numerals, where N is a numeral. Of course, many different associations are possible and the claimed subject matter is intended to cover all such associations regardless of the particular embodiment. Thus, according to particular association embodiments, for example, three valued edge label trees may be expressed as numerals, four valued edge labeled trees may be expressed as numerals and so forth. Thus, manipulations, such as those previously described, for example, in aforementioned U.S. patent application Ser. No. 11/005,859, as well as additional manipulations, may be applied to N valued edge labeled trees, as described in U.S. patent application Ser. No. 11/006,446, filed on Dec. 6, 2004 by J. J, LeTourneau, titled “Method and/or System for Tagging Trees,” and assigned to the assignee of the presently claimed subject matter. However, these are merely examples of how edge labeled trees may be manipulated and the claimed subject matter is not limited in this respect.

According to an embodiment, a “set” may define a collection of unique unordered elements or members. For example, for the purpose of illustration, a set may contain members or elements of a distinct classification, such as the set of all automobiles in a parking lot of a particular color. In a particular embodiment illustrated in FIG. 6, for example, trees may be classified into sets defined by certain characteristics, such as the particular view expressed, and the like. However, this is merely an example of how trees may be classified into distinct sets and the claimed subject matter is not limited in this respect. As illustrated below, symbol strings may similarly be classified based, at least in part, on the particular view that may be expressed in such symbol strings.

According to an embodiment, a “transformation” may define a process by which elements of a first set may be mapped between and/or associated with elements of a second set. For example, in a particular embodiment, a transformation may map and/or associate a member of the set of trees with a member of the set of symbol strings. Similarly, according to a particular embodiment, a transformation may map and/or associate a member of the set of natural numerals to a member in either the set of trees or the set of symbol strings. However, these are merely examples of how a transformation may map and/or associate members of one set to members of another set, and the claimed subject matter is not limited in these respects.

According to an embodiment, a transformation may associate and/or map an element in a first set with an element in a second set as an “elementary equivalent.” Here, in a particular embodiment, such a transformation may associate such elementary equivalent elements in different sets in a one-to-one and onto mapping in which the transformation defines for elements in a first set corresponding, unique elementary equivalents in a second set. Similarly, a transformation may also map and/or associate members of the second set to or between corresponding elementary equivalents in the first set. Examples of a transformations mapping and/or associating elementary equivalent elements of different sets may be found in, for example, Logical Number Theory, An Introduction, by Craig Smorynski, p. 305. However, this is merely an example of how a transformation may map and/or associate members of different sets as elementary equivalents, and the claimed subject matter is not limited in these respects.

FIGS. 7 through 11 are tables illustrating strings of grouping symbols according to an embodiment. As discussed in the aforementioned U.S. patent application Ser. No. 11/005,859, strings may be associated with numerals according to particular association embodiments. Grouping symbols may comprise symbols in a string of symbols that define relationships or groupings among symbols in the string. Particular embodiments relate to pairs of such grouping symbols. However, grouping symbols are not limited to grouping symbols in such grouping symbol pairs, and that grouping symbols may comprise grouping symbol triplets, quadruplets, etc.

As illustrated in FIGS. 7 through 11, in this particular embodiment, such strings may be formed from grouping symbol pairs to obey rules of parenthetical association. For example, in this particular case, including an “open symbol” (e.g., “(,” “[” or “{”, in particular embodiments) for every corresponding “close symbol” (e.g., “),” “]” or “{”) and including a close symbol for every corresponding open symbol. The symbol strings may also be formed with symbol character pairs indicating a “close” or “end” for a particular string and/or portion of a string (e.g., in a particular embodiment, “{([( )])} ( )” and not “{([(]))}”). As in one example, if a first open symbol of a first type of grouping symbol pair is followed by a second open symbol of a second type of grouping symbol pair, a close symbol corresponding to the first open symbol may follow a close symbol corresponding to the second open symbol for proper parenthetical association. Symbol strings comprising grouping symbol pairs obeying these rules of parenthetical association are referred to herein as “well formed” symbol strings. However, these are merely examples of how strings may be formed from character symbols and the claimed subject matter is not limited in these respects.

The embodiments of FIGS. 7 through 11 show grouping symbols which are selected from characters set forth by the American Standard Code for Information Interchange (ASCII). However, these are mere examples of grouping symbols that may be used to represent information in a symbol string and the claimed subject matter is not limited in these respects.

Additionally, as with trees, as described above, such strings of grouping symbols may also define monadic and dyadic operations. For example, combining symbol strings end-to-end may define a dyadic “concatenation” operation in which a numeral representing the resulting combined string, according to an association embodiment, may comprise the product of numerals representing respective combined symbol strings. Also, encapsulating a symbol string with corresponding open and close symbols of a grouping symbol pair may define a monadic “encapsulation” operation. Here, in a particular embodiment as illustrated in FIGS. 7 through 11, an encapsulation of a symbol string “xxxxxx” may be represented as “(xxxxxx)”, “[xxxxxx]” and/or “{xxxxxx}”. However, these are merely examples of operations which may be used to manipulate a symbol string and the claimed subject matter is not limited to the use of these particular operations.

FIGS. 7 through 11 illustrate corresponding association embodiments associating symbol strings with corresponding numerals zero through ten. The symbol “λ” is used to represent the empty symbol string comprising no symbols, but is associated with a single node tree in this particular embodiment. It should be observed that in these particular association embodiments, the single node tree is associated with the numeral zero. However, as illustrated above, other association embodiments may associate a single node tree with numeral one. Particular association embodiments may define associations of one or more non-composite numerals of an initial sequence of non-composite numerals with a symbol string comprising one or more grouping symbol pairs. For example, numeral one is associated with “( )” in association embodiments 700, 1000 and 1100. Numeral one is associated with “{ }” in association embodiment 800 and “[ ]” in association embodiment 900. Numeral two is associated with “( )( )” in association embodiments 700, 1000 and 1100. Numeral two is associated with “{ } { }” in association embodiment 800 and “[ ] [ ]” in association embodiment 900. Numeral three is associated with “{ }” in embodiments 1000 and 1100. Numeral five is associated with “[ ]” in embodiment 1100. It should be understood, however, that these are merely examples of symbols that may represent initial non-composite numerals according to association embodiments and that other embodiments within the scope of the claimed subject matter may employ different symbols. As illustrated below, character strings representing remaining natural numerals of an association embodiment may be generated using the aforementioned concatenation and encapsulation operations.

As pointed out above, according to a particular embodiment, in general a concatenation of symbol strings for a concatenation operation results in a symbol string associated with a numeral that is a product of numerals representing respective concatenated symbol strings. However, an exception is that in the particular embodiments illustrated in FIGS. 7 through 11, multiplication of a numeral associated with a symbol string by a factor of two may be represented by the concatenation of a single pair of open and close symbols (e.g., “( )” for embodiments 700, 1000 and 1100, “{ }” for embodiment 800 and “[ ]” for embodiment 900) which in this embodiment represents a tree associated with numeral one. Thus, in these particular association embodiments, multiplication by a power of two, 2^(n), may be expressed as a concatenation of a symbol string comprising a sequence of n such grouping symbol pairs. However, this is merely an example of how a multiplication of numerals by a power of two may be represented as a concatenation of symbol strings and the claimed subject matter is not limited in these respects.

According to embodiment 700 of FIG. 7, an encapsulation operation may comprise providing a symbol string “(xxxxxx)” that is associated with the numeral Q(n) where n is the numeral associated with the symbol string “xxxxxx” according to this particular association embodiment. Accordingly, the symbol string associated with numeral three comprises “(( ))”, an encapsulation of the symbol representing the numeral one “( ).” The character string representing the numeral six may then be generated by concatenating the symbol strings “( )” (representing multiplication by a factor of two) and “(( ))”, and the symbol string representing the numeral nine may be generated by concatenating the symbol strings “(( ))” and “(( )).” An encapsulation of the symbol string associated with numeral two “( )( )” generates a symbol string “(( )( ))” associated with the numeral five. Concatenating the symbol string “( )” then provides a symbol string “( )(( )( ))” associated with the numeral ten. An encapsulation of the symbol string associated with numeral three “(( ))” generates a symbol string “((( )))” associated with the numeral seven. As should be observed, other than the symbol representing numerals one and two, symbol strings associated with non-composite natural numerals may be generated using the encapsulation operation. It should be evident by using the encapsulation and concatenation operations, symbol strings associated with remaining natural numerals may be generated in like fashion. However, these are merely examples of how the aforementioned concatenation and encapsulation operations may be used for generating symbol strings representing numerals and claimed subject matter is not limited in this respect.

It should be evident from the association of symbol strings and numerals illustrated above in embodiment 700 that symbol strings comprising different types of grouping symbols associated with numerals may similarly be generated for embodiments 800 and 900. Here, for example, multiplication of a numeral associated with a symbol string may be represented by concatenating grouping symbol pairs “{ }” or “[ ]” in embodiments 800 and 900, respectively. Also, by defining the symbol strings representing the numeral one (“{ }” in embodiment 800 and “[ ]” in embodiment 900) and two (“{ } { }” in embodiment 800 and “[ ] [ ]” in embodiment 900), symbol strings representing remaining non-composite numerals may be generated using encapsulation operations. Accordingly, concatenation and encapsulation operations in respective embodiments 700, 800 and 900 may be used to generate symbol strings associated with remaining natural numerals. Again, however, these are merely examples of how the aforementioned concatenation and encapsulation operations may be used for generating symbol strings representing numerals and claimed subject matter is not limited in this respect.

The particular embodiments illustrated with reference to FIGS. 7 through 9 are capable of expressing single views of a tree (e.g., trees having unlabeled nodes and/or edges or “unary” trees). Here, for example, symbol strings may comprise a single type of grouping symbol pair to express a single view while multiple types of grouping symbols may enable the expression of multiple views. In this embodiment, operations applied to symbol strings may express a push operation applied to trees and/or merger of trees at a root node, the former by encapsulating the symbol string and the later by concatenating symbol strings associated with the merged trees. Accordingly, the individual embodiments of FIGS. 7 through 9 may be capable of expressing unary trees as a symbol string comprising a single type of grouping symbol pairs. Again, as shall be made clear, the use of multiple types of grouping symbols may enable the expression of multiple views. As illustrated below with reference to FIGS. 10 and 11, for example, the use of two types of grouping symbol pairs (e.g., in one particular embodiment, “( )” and “{ }”) in a symbol string enables the expression of both views of a binary tree and the use of three types of grouping symbols (e.g., in another particular embodiment, “( ),” “{ }” and “[ ]”) in a symbol string enables the expression of all three views of a tertiary tree. Accordingly, by having N types of grouping symbol, a symbol string may express a single view and up to an N number of views.

FIG. 10 is a table illustrating an association of numerals symbol strings comprising grouping symbol pairs “( ),” and “{ }” which enables the expression of up to two views. Here, the aforementioned concatenation operation of embodiment 700 also applies in this particular association embodiment. That is, by this convention, multiplication of a numeral associated with a symbol string by a factor of two is represented by a concatenation of the symbol string “( ).” Multiplication of numerals associated with other symbol strings may be represented by a concatenation of the symbol strings.

To generate symbol strings associated with non-composite numerals, embodiment 1000 defines two encapsulation operations corresponding with the grouping symbol pairs “( )” and “{ }.” In this particular association embodiment, as pointed out above, symbol strings associated with the first three non-composite numerals, one, two and three, comprise “( ),” “( )( )” and “{ },” respectively. Again, these are merely examples of symbol strings that may be selected as being associated with numerals one, two and three according to a particular association embodiment, and the claimed subject matter is not limited in these respects. As illustrated in FIG. 10, the symbol string associated with numeral three comprises “},” the symbol string associate with the next non-composite numeral, five, comprises “(( ))” and the symbol string associated with the next non-composite numeral, seven, comprises “{( )}.” Accordingly, for this particular embodiment, an encapsulation operation on a symbol string associated with a numeral n provides a symbol string associated with a numeral may be defined as follows:

-   -   Q(2n)=numeral associated with an encapsulation by “( )” of a         symbol string associated with numeral n (where n is a natural         numeral greater than 0); and     -   Q(2n+1)=numeral associated with an encapsulation by “{ }” of a         symbol string associated with numeral n (where n is a natural         numeral greater than or equal to 0).

According to an embodiment, the association embodiment 1000 of FIG. 10 may be used to express a view two tree (e.g., a BELT) as a single string of grouping symbol pairs. In this particular embodiment, “1” and “0” are used to express two views but other distinct labels may be used to express two views. Here, in a particular embodiment, a single node tree may be associated with the grouping symbol pair “( )”, which is associated with numeral one. As illustrated in FIG. 12, a single node connected by an edge having a zero label value may be represented as “( )” and a single node connected by an edge having a one label may be represented as “{ }.” Encapsulation by “( )” of the symbol string associated with tree 1206 provides a symbol string “({ })” associated with tree 1208 and corresponding to a zero-push operation on tree 1206. Similarly, encapsulation by “{ }” of the symbol string associated with tree 1202 provides a symbol string “{( )}” which is associated with tree 1210 and corresponding to a one-push operation on tree 1202. Accordingly, there is a one-to-one correspondence between push operations on BELTs and encapsulation operations on corresponding symbol strings.

Tree 1202, associated with symbol string “( )” and numeral one in the presently illustrated association embodiment, may be merged with itself to form tree 1204 which is associated with the numeral two and a concatenation of the grouping symbol pair “( )” and symbol string “( )( ).” Accordingly, concatenation of the symbol pair “( )” to a symbol string corresponds with a merger of tree 1202 with a tree corresponding to the symbol string. Similarly, tree 1206 associated with grouping symbol pair “{ }” may be merged with tree 1210 associated with symbol string “{( )}” to provide tree 1212. Here, the resulting tree 1212 is associated with the concatenation of the grouping symbol pair “{ }” and symbol string “{( )}.” Accordingly, there is a one-to-one correspondence between merger operations on BELTs and concatenation operations on symbol strings in the presently illustrated association embodiment.

Using the same association of BELTs and symbol strings comprising grouping symbol pairs “( )” and “{ }” illustrated above with reference to FIG. 12, applying concatenation operations to symbol strings to represent associated tree merger operations on trees and applying encapsulation operations to symbol strings to represent associated push operations, any BELT may be transformed to a string of grouping symbol pairs. An additional example of such a transformation is shown in FIG. 13. As illustrated in this particular association embodiment, there exists a corresponding symbol string for any BELT and there exists a corresponding BELT for every symbol string formed according to embodiment 1000. Accordingly, such BELTs and symbol strings comprise elementary equivalents.

FIG. 11 shows a schematic diagram of an association of numerals and symbol strings comprising symbol pairs “( ),” “{ },” and “[ ].” Here, the aforementioned concatenation operation of embodiment 700 also applies. That is, multiplication of a numeral associated with a symbol string by a factor of two is represented by a concatenation of the symbol string “( ).” Multiplication of numerals associated with other symbol strings may be represented by a concatenation of the symbol strings.

To generate symbol strings associated with non-composite numerals, embodiment 1100 defines three encapsulation operations corresponding with the encapsulation symbol pairs “( ),” “{ }” and “[ ].” In this particular association embodiment, symbol strings associated with the first four non-composite numerals, one, two, three and five comprise “( )”, “( )( )” “{ }” and “[ ],” respectively. However, these are merely examples of symbol strings that may be selected as being associated with the first four non-composite numerals and the claimed subject matter is not limited in these respects. Three encapsulation operations may be used to generate symbol strings corresponding to non-composite numerals of seven or greater in an alternating fashion. As illustrated in FIG. 11, the symbol strings associated with the non-composite numerals seven, eleven and thirteen comprise “(( )),” “{( )}” and “[( )],” respectively. Accordingly, for this particular embodiment, encapsulation operation on a symbol string associated with a numeral n provides a symbol string associated with a numeral may be defined as follows:

-   -   Q(3n)=numeral associated with an encapsulation by “( )” of a         symbol string associated with numeral n (where n is a natural         numeral greater than 0);     -   Q(3n+1)=numeral associated with an encapsulation by “{ }” of a         symbol string associated with numeral n (where n is a natural         numeral greater than or equal to 0); and     -   Q(3n+2)=numeral associated with an encapsulation by “[ ]” of a         symbol string associated with numeral n (where n is a natural         numeral greater than or equal to 0).

According to an embodiment, the association embodiment 1100 of FIG. 11 may be used to express a view three tree as a single string of grouping symbol pairs. Here, in this particular embodiment, by convention, a single node tree may be associated with the grouping symbol pair “( )”, which is associated with numeral one. FIG. 14 is a schematic diagram illustrating an association of tertiary trees and symbol strings according to an association embodiment. In this particular embodiment, edges may be associated with a label value of A, B or C. However, these are merely examples of labels that may be used to express three views and that any other distinguishable three labels may be used. Here, a single node connected by an edge having a label value A may be represented as “( ),” a single node connected by an edge having a label value B may be represented as “{ }” and a single node connected by an edge having a label value C may be represented as “[ ].” Encapsulation by “( )” of the symbol string associated with tree 1404 provides a symbol string “({ })” associated with tree 1208 and corresponding to an A-push operation on tree 1412. Similarly, encapsulation by “{ }” of the symbol string associated with tree 1404 provides a symbol string “{( )}” associated with tree 1430 and corresponding to a B-push operation on tree 1404. In yet another example, encapsulation by “[ ]” of the symbol string associated with tree 1406 provides a symbol string “[{ }]” associated with tree 1440 and corresponding to a C-push operation on tree 1440. Accordingly, there is a one-to-one correspondence between push operations on A, B, C edge labeled tertiary trees and encapsulation operations on corresponding symbol strings.

Tree 1404, associated with symbol string “( )” and numeral one in the presently illustrated association embodiment, may be merged with itself to form tree 1410 which is associated with the numeral two and a concatenation of the grouping symbol pair “( )” with itself. Accordingly, concatenation of the symbol pair “( )” to a symbol string corresponds with a merger of tree 1404 with a tree corresponding to the symbol string. Similarly, tree 1406 associated with grouping symbol pair “{ }” may be merged with tree 1430 associated with symbol string “{( )}” to provide tree 1414. Here, the resulting tree 1414 is associated with a symbol string comprising a concatenation of the grouping symbol pair “{ }” and symbol string “{( )}.” In another example, tree 1408 associated with grouping symbol pair “[ ]” may be merged with tree 1412 associated with symbol string “({ })” to provide tree 1416. Here, the resulting tree 1416 is associated with a symbol string comprising a concatenation of the grouping symbol pair “[ ]” and symbol string “({ }).” Accordingly, there is a one-to-one correspondence between merger operations on A, B, C edge labeled tertiary trees and concatenation operations on symbol strings in the presently illustrated association embodiment.

Using the same association of A, B, C edge labeled tertiary trees and symbol strings comprising grouping symbol pairs “( ),” “{ }” and “[ ]” illustrated above with reference to FIG. 14, applying concatenation operations on symbol strings to represent associated merger operations on trees and applying encapsulation operations on symbol strings to represent associated push operations, any A,B,C edge labeled tertiary tree may be transformed to a string of grouping symbol pairs and vice-versa. An additional example of such a transformation is shown in FIG. 15. As illustrated in this particular association embodiment 1100, as illustrated above with BELTs in embodiment 1000, there exists a corresponding symbol string for any A,B,C edge labeled tertiary tree and there exists a corresponding A,B,C edge labeled tertiary tree for every symbol string formed according to this particular association embodiment. Accordingly, such A,B,C edge labeled trees and symbol strings comprise elementary equivalents.

As discussed above operations and/or manipulations applied to symbol strings may be related to “arithmetic operations” applied to the numerals that correspond with the manipulated symbol strings. One example above, in a particular embodiment, illustrates that the dyadic concatenation operation applied to symbol strings relates to a dyadic multiplication operation on numerals representing the concatenated symbol strings. Another example above, according to a particular embodiment, illustrates that the monadic encapsulation operation on symbol strings relates to a monadic operation applied to numerals representing the encapsulated symbol strings to provide corresponding non-composite numerals. However, these are merely examples of arithmetic operations that may correspond with manipulations of and/or operations applied to symbol strings, and claimed subject matter may include other such arithmetic operations.

As discussed above in connection with embodiments 1000 and 1100, for a symbol string formed according to the corresponding association embodiment there exist a unique numeral and/or tree (here, a BELT or labeled tertiary tree in these particular examples). Accordingly, these relationships and/or associations among symbol strings, numerals and/or trees provide various associations between numerals and trees. Here, numerals associated with symbol strings in association embodiments 1000 and 1100 may also be associated with the trees associated with the symbol strings. Therefore, these numerals and trees associated with the symbol strings themselves form an associations of trees and numerals according to particular association embodiments.

The embodiments of FIGS. 10 and 11 illustrate that using multiple types of grouping symbol pairs, a symbol string may represent multiple views of a tree. Also, the embodiments of FIGS. 7 through 11 employ arbitrarily chosen grouping symbol pairs. It should be understood that other grouping symbol pairs may be formed from character pairs (ASCII or otherwise) that distinguish between open and close symbols such as, for example, “< >,” “\/,” “$ %,” “! @,” “T t,” “A a,” etc., and that claimed subject matter is not limited in this respect. Also, it should be understood that such symbols may comprise more than one character if desired. Referring to the embodiments of FIGS. 4 and 5, for example, an encapsulation operation may be expressed using an open symbol “A(”, “B(”, “C(” or “D(” comprising two characters and followed by a close symbol “)”.

Also, it should be understood that the grouping symbol pairs employed in the particular embodiments illustrated herein are chosen to be perceivable by human inspection for the purposes of illustrating particular examples of the claimed subject matter. However, it should be understood that the claimed subject matter is also directed to the use of grouping symbols that are not perceivable by human inspection including, for example, grouping symbols that are encoded in magnetic, optical, semiconductor and/or radio frequency energy mediums, or any other type of media capable of expressing information.

While particular embodiments illustrated herein show particular types of label values as being associated with edges connecting nodes, edge label values for other embodiments may not comprise simply natural numerals, such as, as one example, trees that include negative numerals, such data labels may be converted to an ordered pair of numerals. For example, the first numeral may represent a data type. Examples include a data type such as negative, dollars, etc. As described above, such trees may also be converted to edge labeled trees, for example. However, again, this is provided for purposes of explanation and/or illustration. The claimed subject matter is not limited in scope to employing the approach of the previously referenced provisional patent application.

Thus, according to an embodiment, it may be desirable to transform between a tree and a symbol string for storing in a storage medium and/or processing in a computer platform, for example. If such a tree has a particular view, the symbols of such a string may also be chosen to express information for that view. In one embodiment, symbol strings comprising grouping symbol pairs such as “( )”, “{ }” and “[ ]” may express trees as previously described. In the embodiment of FIG. 5, for example, encapsulation of a symbol string representing a tree by a grouping symbol pair “( )”, “{ }” or “[ ]” may provide a symbol string representing an A-push, B-push or C-push applied to the tree. Accordingly, multiple views of a tree may be expressed by using corresponding multiple types of grouping symbol pairs. A merger of trees at a root node may be expressed by concatenating symbol strings representing the merged trees.

Particular embodiments distinguish among grouping symbol pair “types” to express corresponding multiple views of trees. In the embodiments of FIGS. 7 through 9, for example, a single grouping symbol type is used to express trees of view one. In the embodiment of FIG. 10, employing two grouping symbol pair types “( )” and “{ }”, on the other hand, it is possible to express trees of view two. In the embodiment of FIG. 11, employing three grouping symbol pair types “( )”, “{ }” and “[ ]”, it is possible to express trees of view three. Other embodiments within the scope of the claimed subject matter may also be directed to the use of four or more grouping symbol pair types to express views of higher value trees.

It will, of course, be understood that, although particular embodiments have just been described, the claimed subject matter is not limited in scope to a particular embodiment or implementation. For example, one embodiment may be in hardware, such as implemented to operate on a device or combination of devices, for example, whereas another embodiment may be in software. Likewise, an embodiment may be implemented in firmware, or as any combination of hardware, software, and/or firmware, for example. Likewise, although the claimed subject matter is not limited in scope in this respect, one embodiment may comprise one or more articles, such as a storage medium or storage media. This storage media, such as, one or more CD-ROMs and/or disks, for example, may have stored thereon instructions, that when executed by a system, such as a computer system, computing platform, or other system, for example, may result in an embodiment of a method in accordance with the claimed subject matter being executed, such as one of the embodiments previously described, for example. As one potential example, a computing platform may include one or more processing units or processors, one or more inpuvoutput devices, such as a display, a keyboard and/or a mouse, and/or one or more memories, such as static random access memory, dynamic random access memory, flash memory, and/or a hard drive. For example, a display may be employed to display one or more queries, such as those that may be interrelated, and or one or more tree expressions, although, again, the claimed subject matter is not limited in scope to this example.

In the preceding description, various aspects of the claimed subject matter have been described. For purposes of explanation, specific numbers, systems and/or configurations were set forth to provide a thorough understanding of the claimed subject matter. However, it should be apparent to one skilled in the art having the benefit of this disclosure that the claimed subject matter may be practiced without the specific details. In other instances, well-known features were omitted and/or simplified so as not to obscure the claimed subject matter. While certain features have been illustrated and/or described herein, many modifications, substitutions, changes and/or equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and/or changes as fall within the true spirit of the claimed subject matter. 

1. A method comprising: transforming one or more trees to one or more corresponding symbol strings according to an association of trees and symbol strings comprising up to two or more grouping symbol types, said one or more symbol strings being associated with one or more corresponding numerals according to an association of symbol strings and numerals; and manipulating said one or more symbol strings to provide a resulting symbol string corresponding with a resulting numeral according to said association of symbol strings and numerals, wherein said resulting numeral is representative of an arithmetic operation on said one or more corresponding numerals.
 2. The method of claim 1, wherein said manipulating said one or more symbol strings further comprises concatenating a first symbol string with a second symbol string, said first and second symbol strings being associated with corresponding first and second numeral according to the association of strings and numerals, to provide a concatenated symbol string associated with the resulting numeral according to the association of strings and numerals comprising a product of said first and second numerals.
 3. The method of claim 2, wherein said first symbol string is associated with a first tree and said second symbol string is associated with a second tree according to the association of symbol strings and trees, and wherein said concatenated symbol string is associated with a merger of said first and second trees according to said association of symbol strings and trees.
 4. The method of claim 1, wherein at least one of said one or more symbol strings comprise grouping symbol pairs comprising corresponding open symbol and close symbols, and wherein said manipulating said one or more symbol strings further comprises encapsulating said at least one symbol string with open and close symbols of a symbol pair to represent a push operation on a tree associated with said at least one symbol string according to said association of symbol strings and trees.
 5. The method of claim 4, wherein the encapsulated symbol string is associated with a non-composite numeral according to the association of symbol strings and numerals.
 6. The method of claim 5, wherein the non-composite numeral is associated with a tree resulting from said push operation according to an association of trees and numerals.
 7. The method of claim 1, wherein at least one of said trees expresses two or more views corresponding to said two more types of grouping symbols.
 8. The method of claim 7, wherein said grouping symbols comprise grouping symbol pairs comprising corresponding open symbols and close symbols, and wherein said manipulating said one or more symbol strings further comprises encapsulating at least one of said symbol strings with open and close symbols of a grouping symbol pair of a first type of grouping symbol pairs to represent a push operation on a tree corresponding with said at least one of said symbol strings according to said association of trees and symbol strings, the push operation being associated with a first view corresponding with the first type of grouping symbol pair.
 9. The method of claim 8, wherein said manipulating said one or more symbol strings further comprises encapsulating said encapsulated symbol string with open and close symbols of a grouping symbol pair of a second type of grouping symbol pairs to represent a second push operation on said pushed tree, the second push operation being associated with a second view corresponding with the second type of grouping symbol pair.
 10. The method of claim 9, wherein the said tree comprises a binary edge labeled tree, and wherein the push operation associated with the first view comprises a zero-push operation and the push operation associated with the second view comprises a one-push operation.
 11. The method of claim 1, wherein the grouping symbols comprise open and close symbols selected from among ASCII characters.
 12. An apparatus comprising: means for transforming one or more trees to one or more corresponding symbol strings according to an association of trees and symbol strings comprising up to two or more grouping symbol types, said one or more symbol strings being associated with one or more corresponding numerals according to an association of symbol strings and numerals; and means for manipulating said one or more symbol strings to provide a resulting symbol string corresponding with a resulting numeral according to said association of symbol strings and numerals, wherein said resulting numeral is representative of an arithmetic operation on said one or more corresponding numerals.
 13. The apparatus of claim 12, wherein said means for manipulating said one or more symbol strings further comprises means for concatenating a first symbol string with a second symbol string, said first and second symbol strings being associated with corresponding first and second numeral according to the association of strings and numerals, to provide a concatenated symbol string associated with the resulting numeral according to the association of strings and numerals comprising a product of said first and second numerals.
 14. The apparatus of claim 13, wherein said first symbol string is associated with a first tree and said second symbol string is associated with a second tree according to the association of symbol strings and trees, and wherein said concatenated symbol string is associated with a merger of said first and second trees according to said association of symbol strings and trees.
 15. The apparatus of claim 12, wherein at least one of said one or more symbol strings comprise grouping symbol pairs comprising corresponding open symbol and close symbols, and wherein said means for manipulating said one or more symbol strings further comprises means for encapsulating said at least one symbol string with open and close symbols of a symbol pair to represent a push operation on a tree associated with said at least one symbol string according to said association of symbol strings and trees.
 16. The apparatus of claim 15, wherein the encapsulated symbol string is associated with a non-composite numeral according to the association of symbol strings and numerals.
 17. The apparatus of claim 16, wherein the non-composite numeral is associated with a tree resulting from said push operation according to an association of trees and numerals.
 18. The apparatus of claim 12, wherein at least one of said trees expresses two or more views corresponding to said two more types of grouping symbols.
 19. The apparatus of claim 18, wherein said grouping symbols comprise grouping symbol pairs comprising corresponding open symbols and close symbols, and wherein said means for manipulating said one or more symbol strings further comprises means for encapsulating at least one of said symbol strings with open and close symbols of a grouping symbol pair of a first type of grouping symbol pairs to represent a push operation on a tree corresponding with said at least one of said symbol strings according to said association of trees and symbol strings, the push operation being associated with a first view corresponding with the first type of grouping symbol pair.
 20. The apparatus of claim 19, wherein said means for manipulating said one or more symbol strings further comprises means for encapsulating said encapsulated symbol string with open and close symbols of a grouping symbol pair of a second type of grouping symbol pairs to represent a second push operation on said pushed tree, the second push operation being associated with a second view corresponding with the second type of grouping symbol pair
 21. The apparatus of claim 20, wherein the said tree comprises a binary edge labeled tree, and wherein the push operation associated with the first view comprises a zero-push operation and the push operation associated with the second view comprises a one-push operation.
 22. The apparatus of claim 12, wherein the grouping symbols comprise open and close symbols selected from among ASCII characters.
 23. An article comprising: a storage medium comprising machine-readable stored thereon to: transform one or more trees to one or more corresponding symbol strings according to an association of trees and symbol strings comprising up to two or more grouping symbols types, said one or more symbol strings being associated with one or more corresponding numerals according to an association of symbol strings and numerals; and manipulate said one or more symbol strings to provide a resulting symbol string corresponding with a resulting numeral according to said association of symbol strings and numerals, wherein said resulting numeral is representative of an arithmetic operation on said one or more corresponding numerals.
 24. The article of claim 23, wherein said storage medium further comprises machine-readable instructions stored thereon to concatenate a first symbol string with a second symbol string, said first and second symbol strings being associated with corresponding first and second numeral according to the association of strings and numerals, to provide a concatenated symbol string associated with the resulting numeral according to the association of strings and numerals comprising a product of said first and second numerals.
 25. The article of claim 24, wherein said first symbol string is associated with a first tree and said second symbol string is associated with a second tree according to the association of symbol strings and trees, and wherein said concatenated symbol string is associated with a merger of said first and second trees according to said association of symbol strings and trees.
 26. The article of claim 23, wherein at least one of said one or more symbol strings comprise grouping symbol pairs comprising corresponding open symbol and close symbols, and wherein the storage medium further comprises machine-readable instructions stored thereon to encapsulate said at least one symbol string with open and close symbols of a symbol pair to represent a push operation on a tree associated with said at least one symbol string according to said association of symbol strings and trees.
 27. The article of claim 26, wherein the encapsulated symbol string is associated with a non-composite numeral according to the association of symbol strings and numerals.
 28. The article of claim 27, wherein the non-composite numeral is associated with a tree resulting from said push operation according to an association of trees and numerals.
 29. The article of claim 23, wherein at least one of said trees expresses two or more views corresponding to said two more types of grouping symbols.
 30. The article of claim 29, wherein said grouping symbols comprise grouping symbol pairs comprising corresponding open symbols and close symbols, and wherein said storage medium further comprises machine-readable instructions stored thereon to encapsulate at least one of said symbol strings with open and close symbols of a grouping symbol pair of a first type of grouping symbol pairs to represent a push operation on a tree corresponding with said at least one of said symbol strings according to said association of trees and symbol strings, the push operation being associated with a first view corresponding with the first type of grouping symbol pair.
 31. The article of claim 30, wherein said storage medium further comprises machine-readable instructions stored thereon to encapsulate said encapsulated symbol string with open and close symbols of a grouping symbol pair of a second type of grouping symbol pairs to represent a second push operation on said pushed tree, the second push operation being associated with a second view corresponding with the second type of grouping symbol pair
 32. The article of claim 31, wherein the said tree comprises a binary edge labeled tree, and wherein the push operation associated with the first view comprises a zero-push operation and the push operation associated with the second view comprises a one-push operation.
 33. The article of claim 23, wherein the grouping symbols comprise open and close symbols selected from among ASCII characters.
 34. An apparatus comprising a computing platform, the computing platform being adapted to: transform one or more trees to one or more corresponding symbol strings according to an association of trees and symbol strings comprising up to two or more grouping symbols types, said one or more symbol strings being associated with one or more corresponding numerals according to an association of symbol strings and numerals; and manipulate said one or more symbol strings to provide a resulting symbol string corresponding with a resulting numeral according to said association of symbol strings and numerals, wherein said resulting numeral is representative of an arithmetic operation on said one or more corresponding numerals.
 35. The apparatus of claim 34, wherein the computing platform is further adapted to concatenate a first symbol string with a second symbol string, said first and second symbol strings being associated with corresponding first and second numeral according to the association of strings and numerals, to provide a concatenated symbol string associated with the resulting numeral according to the association of strings and numerals comprising a product of said first and second numerals.
 36. The apparatus of claim 35, wherein said first symbol string is associated with a first tree and said second symbol string is associated with a second tree according to the association of symbol strings and trees, and wherein said concatenated symbol string is associated with a merger of said first and second trees according to said association of symbol strings and trees.
 37. The apparatus of claim 34, wherein at least one of said one or more symbol strings comprise grouping symbol pairs comprising corresponding open symbol and close symbols, and wherein said computing platform is further adapted to encapsulate said at least one symbol string with open and close symbols of a symbol pair to represent a push operation on a tree associated with said at least one symbol string according to said association of symbol strings and trees.
 38. The apparatus of claim 37, wherein the encapsulated symbol string is associated with a non-composite numeral according to the association of symbol strings and numerals.
 39. The apparatus of claim 38, wherein the non-composite numeral is associated with a tree resulting from said push operation according to an association of trees and numerals.
 40. The apparatus of claim 34, wherein at least one of said trees expresses two or more views corresponding to said two more types of grouping symbols.
 41. The apparatus of claim 40, wherein said grouping symbols comprise grouping symbol pairs comprising corresponding open symbols and close symbols, and wherein said computing platform is further adapted to encapsulate at least one of said symbol strings with open and close symbols of a grouping symbol pair of a first type of grouping symbol pairs to represent a push operation on a tree corresponding with said at least one of said symbol strings according to said association of trees and symbol strings, the push operation being associated with a first view corresponding with the first type of grouping symbol pair.
 42. The apparatus of claim 41, wherein said computing platform is further adapted to encapsulate said encapsulated symbol string with open and close symbols of a grouping symbol pair of a second type of grouping symbol pairs to represent a second push operation on said pushed tree, the second push operation being associated with a second view corresponding with the second type of grouping symbol pair
 43. The apparatus of claim 42, wherein the said tree comprises a binary edge labeled tree, and wherein the push operation associated with the first view comprises a zero-push operation and the push operation associated with the second view comprises a one-push operation.
 44. The apparatus of claim 34, wherein the grouping symbols comprise open and close symbols selected from among ASCII characters.
 45. An article comprising: a storage medium comprising machine-readable information stored thereon representative of a resulting symbol string comprising grouping symbols, said grouping symbols being determined, at least in part, on: a transformation of one or more trees to one or more corresponding symbol strings according to an association of trees and symbol strings comprising up to two or more grouping symbols types, said one or more symbol strings being associated with one or more corresponding numerals according to an association of symbol strings and numerals; and a manipulation of said one or more symbol strings to provide said resulting symbol string corresponding with a resulting numeral according to said association of symbol strings and numerals, wherein said resulting numeral is representative of an arithmetic operation on said one or more corresponding numerals.
 46. The article of claim 45, wherein said resulting symbol string comprises a concatenation of a first symbol string with a second symbol string, said first and second symbol strings being associated with corresponding first and second numerals according to the association of strings and numerals, said resulting numeral comprising a product of said first and second numerals.
 47. The article of claim 46, wherein said first symbol string is associated with a first tree and said second symbol string is associated with a second tree according to the association of symbol strings and trees, and wherein said concatenated symbol string is associated with a merger of said first and second trees according to said association of symbol strings and trees.
 48. The article of claim 45, wherein at least one of said one or more symbol strings comprise grouping symbol pairs comprising corresponding open symbol and close symbols, and wherein said resulting symbol string further comprises an encapsulation of said at least one symbol string with open and close symbols of a symbol pair to represent a push operation on a tree associated with said at least one symbol string according to said association of symbol strings and trees.
 49. The article of claim 48, wherein the encapsulated symbol string is associated with a non-composite numeral according to the association of symbol strings and numerals.
 50. The article of claim 49, wherein the non-composite numeral is associated with a tree resulting from said push operation according to an association of trees and numerals.
 51. The article of claim 45, wherein at least one of said trees expresses two or more views corresponding to said two more types of grouping symbols.
 52. The article of claim 51, wherein said grouping symbols comprise grouping symbol pairs comprising corresponding open symbols and close symbols, and wherein said resulting symbol string further comprises an encapsulation of the at least one of said symbol strings with open and close symbols of a grouping symbol pair of a first type of grouping symbol pairs to represent a push operation on a tree corresponding with said at least one of said symbol strings according to said association of trees and symbol strings, the push operation being associated with a first view corresponding with the first type of grouping symbol pairs.
 53. The article of claim 52, wherein said resulting symbol string further comprises an encapsulation of said encapsulated symbol string with open and close symbols of a grouping symbol pair of a second type of grouping symbol pairs to represent a second push operation on said pushed tree, the second push operation being associated with a second view corresponding with the second type of grouping symbol pairs.
 54. The article of claim 53, wherein the said tree comprises a binary edge labeled tree, and wherein the push operation associated with the first view comprises a zero-push operation and the push operation associated with the second view comprises a one-push operation.
 55. The article of claim 45, wherein the grouping symbols comprise open and close symbols selected from among ASCII characters.
 56. A method comprising: transforming between at least a first and at least a second expression, said expressions having a common view, wherein said common view is other than view one.
 57. The method of claim 56, wherein said expressions respectively include trees and symbol strings.
 58. The method of claim 57, wherein said first expression comprises a tree and said second expression comprises a string.
 59. The method of claim 57, wherein said first expression comprises a symbol string and said second expression comprises a tree.
 60. The method of claim 56, and further comprising, prior to said transforming, applying operations to said first expression.
 61. The method of claim 60, and further comprising, prior to said transforming, applying operations to said second expression.
 62. The method of claim 56, and further comprising, subsequent to said transforming, applying operations to said first expression.
 63. The method of claim 62, and further comprising, subsequent to said transforming, applying operations to said second expression.
 64. The method of claim 56, wherein said first expression comprises one or more numerals and said second expression comprises at least one of a tree and/or a symbol string.
 65. An apparatus comprising: means for representing at least a first expression; and means for transforming between at least a first and at least a second expression, said expressions having a common view, wherein said common view is other than view one.
 66. The apparatus of claim 65, wherein said expressions respectively include trees and symbol strings.
 67. The apparatus of claim 66, wherein said first expression comprises a tree and said second expression comprises a string.
 68. The apparatus of claim 66, wherein said first expression comprises a symbol string and said second expression comprises a tree.
 69. The apparatus of claim 65, and further comprising means for applying operations to said first expression prior to said transforming.
 70. The apparatus of claim 69, and further comprising means for applying operations to said second expression prior to said transforming.
 71. The apparatus of claim 65, and further comprising means for applying operations to said first expression subsequent to said transforming.
 72. The apparatus of claim 71, and further comprising means for applying operations to said second expression subsequent to said transforming.
 73. The apparatus of claim 65, wherein said first expression comprises one or more numerals and said second expression comprises at least one of a tree and/or a symbol string.
 74. An article comprising: a storage medium comprising machine-readable instructions stored thereon to: transform between at least a first and at least a second expression, said expressions having a common view, wherein said common view is other than view one.
 75. The article of claim 74, wherein said expressions respectively include trees and symbol strings.
 76. The article of claim 75, wherein said first expression comprises a tree and said second expression comprises a string.
 77. The article of claim 75, wherein said first expression comprises a symbol string and said second expression comprises a tree.
 78. The article of claim 74, wherein said storage medium further comprises machine-readable instructions stored thereon to apply operations to said first expression prior to said transformation.
 79. The article of claim 78, wherein said storage medium further comprises machine-readable instructions stored thereon to apply operations to said second expression prior to said transformation.
 80. The article of claim 74, wherein said storage medium further comprises machine-readable instructions stored thereon to apply operations to said first expression subsequent to said transforming.
 81. The article of claim 74, wherein said storage medium further comprises machine-readable instructions stored thereon to apply operations to said second expression subsequent to said transforming.
 82. The article of claim 74, wherein said first expression comprises one or more numerals and said second expression comprises at least one of a tree and/or a symbol string.
 83. An apparatus comprising a computing platform, the computing platform being adapted to: transform between at least a first and at least a second expression, said expressions having a common view, wherein said common view is other than view one.
 84. The apparatus of claim 83, wherein said expressions respectively include trees and symbol strings.
 85. The apparatus of claim 84, wherein said first expression comprises a tree and said second expression comprises a string.
 86. The apparatus of claim 84, wherein said first expression comprises a symbol string and said second expression comprises a tree.
 87. The apparatus of claim 83, wherein said computing platform is further adapted to apply operations to said first expression prior to said transformation.
 88. The apparatus of claim 87, wherein said computing platform is further adapted to apply operations to said second expression prior to said transformation.
 89. The apparatus of claim 83, wherein said computing platform is further adapted to apply operations to said first expression subsequent to said transformation.
 90. The apparatus of claim 89, wherein said computing platform is further adapted to apply operations to said second expression subsequent to said transformation.
 91. The article of claim 83, wherein said first expression comprises one or more numerals and said second expression comprises at least one of a tree and/or a symbol string.
 92. An article comprising: a storage medium comprising machine-readable information stored thereon representative of a first expression resulting from transformation from a second expression, said expressions having a common view, wherein said common view is other than view one.
 93. The article of claim 92, wherein said expressions respectively include trees and symbol strings.
 94. The article of claim 93, wherein said first expression comprises a tree and said second expression comprises a string.
 95. The article of claim 93, wherein said first expression comprises a symbol string and said second expression comprises a tree.
 96. The article of claim 92, wherein said first expression comprises one or more numerals and said second expression comprises at least one of a tree and/or a symbol string.
 97. The article of claim 92, wherein said second expression comprises one or more numerals and said first expression comprises at least one of a tree and/or a symbol string. 